Algebra and Geometry with Calculus

One of the earliest algebra
topics learned is how to find the
slope of a line--a numerical value that describes just how slanted that
line is. Calculus gives us a much more generalized method of finding slopes. With
it, we can find not only how steeply a line slopes, but indeed, how steeply any
curve slopes at any given point.

Without calculus, it is difficult to find areas of shapes other than those whose
formulas you learned in geometry. You may be able to find the area of commons shapes
such as a triangle, square, rectangle, circle, and even a trapezoid; but how could
you find the area of the shape like the one shown below?

With calculus, you can calculate complicated x-intercepts. Without a graphing calculator,
it is pretty difficult to calculate an irrational root. However, a simple process
called Newton's Method (named Isaac Newton) allows you to calculate an irrational
root to whatever accuracy you want.

Calculus makes it much easier to visualize graphs. You may already have a good grasp
of linear functions and how to visualize their graphs easily, but what about the
graph of something like y= x^3 + 2x^2 - x + 1? Elementary calculus tells you exactly
where that graph will be increasing, decreasing, and twisting. You can even find
the highest and lowest points on the graph without plotting a single point.

One of the most useful applications of calculus is the optimization of functions.
In a small number of steps, you can answer questions such as:

If I have 500 feet of fence, what is the largest rectangular yard I can make?
or Given a rectangular sheet of paper which measures 8.5 inches by 11 inches, what
are the dimensions of the box I can make containing the greatest volume?

The traditional way to create an open box from a rectangular surface is to cut congruent
squares from the corners of the rectangle and then to fold the resulting sides up
as shown:

Calculus develops concepts in other mathematics that lets us discover more about
them and enables us to achieve greater feats than the mathematics that it is built
on. It is vital to understanding and making sense of the world we live in.

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